ABSTRACT
In analytical calculus, differentiation and integration are intrinsically linked by each being the inverse process of the other. This chapter reviews the various finite difference methods. These methods use the definition of the derivative to estimate the derivative value. The chapter aims to extend these methods to look at the second derivative of a function. Numerical differentiation, like other problems in numerical analysis, begins with the analytical definition of the problem. Newton–Cotes integration forms rely upon the specification of analytical integration to be effective. The midpoint rule, sometimes called the rectangle rule, and the trapezoidal rule are the two most basic examples of Newton–Cotes forms available, representing a one-point interpolation of the curve and a two-point interpolation of the curve, respectively. Like Newton–Cotes integration, Gaussian integration is a weighted sum of function evaluations. Monte Carlo methods have a long history in simulation and other areas of applied mathematics.
